... I'm not sure how to find the real parts when imaginary terms are being crossed with real terms, any ideas?

Re x Re = Re.
Im x Im = (-)Re.
Re x Im = Im.
Im x Re = Im.

You may also use i = e^ipi/2, and add phases to keep the expressions in polar form. In principle, both of these should be possible; however, choosing which way is more convenient comes with experience. Try both, and you will start to develop an intuition for it.

EDIT: Oh, wait, your expression for S is different than what I'm used to. I use Re(ExB*), or actually Re(ExH*). Sorry for the confusion. Anyway, you can't have Re(something) = something x e^ikr.